check the picture below.
so.. "p" is the distance from the vertex to either, the focus point or the directrix.
thus the focus is "p" units from the vertex.
now, if we set the parabola at the origin, like in the picture, and then use another point on the parabola, let's say we'll use (6,1), then
[tex]\bf \textit{parabola vertex form with focus point distance}\\\\
\begin{array}{llll}
(y-{{ k}})^2=4{{ p}}(x-{{ h}}) \\\\
\boxed{(x-{{ h}})^2=4{{ p}}(y-{{ k}})} \\
\end{array}
\qquad
\begin{array}{llll}
vertex\ ({{ h}},{{ k}})\\\\
{{ p}}=\textit{distance from vertex to }\\
\qquad \textit{ focus or directrix}
\end{array}\\\\
-------------------------------\\\\[/tex]
[tex]\bf \begin{cases}
h=0\\
k=0
\end{cases}\implies (x-0)^2=4p(y-0)\implies x^2=4py
\\\\\\
\begin{cases}
x=6\\
y=1
\end{cases}\implies (6)^2=4p(1)\implies \cfrac{6^2}{4}=p\implies \cfrac{36}{4}=p\implies 9=p[/tex]
